Nuprl Lemma : mutual-primitive-recursion
∀[A,B:Type].
∀f0:A. ∀g0:B. ∀F:ℕ ⟶ A ⟶ B ⟶ A. ∀G:ℕ ⟶ A ⟶ B ⟶ B.
∃f:ℕ ⟶ A
∃g:ℕ ⟶ B
(((f 0) = f0 ∈ A)
∧ ((g 0) = g0 ∈ B)
∧ (∀s:ℕ+. (((f s) = F[s - 1;f (s - 1);g (s - 1)] ∈ A) ∧ ((g s) = G[s - 1;f (s - 1);g (s - 1)] ∈ B))))
Proof
Definitions occuring in Statement :
nat_plus: ℕ+,
nat: ℕ,
uall: ∀[x:A]. B[x],
so_apply: x[s1;s2;s3],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
subtract: n - m,
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
member: t ∈ T,
so_apply: x[s1;s2;s3],
subtype_rel: A ⊆r B,
nat: ℕ,
uimplies: b supposing a,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
implies: P ⇒ Q,
prop: ℙ,
top: Top,
so_lambda: λ2x.t[x],
so_apply: x[s],
pi1: fst(t),
pi2: snd(t),
cand: A c∧ B,
nat_plus: ℕ+,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
primrec_wf,
int_seg_subtype_nat,
false_wf,
int_seg_wf,
pi1_wf_top,
equal_wf,
nat_wf,
pi2_wf,
primrec0_lemma,
primrec-unroll,
nat_plus_wf,
and_wf,
le_wf,
all_wf,
nat_plus_subtype_nat,
subtract_wf,
nat_plus_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermSubtract_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_subtract_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
eq_int_wf,
bool_wf,
equal-wf-T-base,
assert_wf,
intformeq_wf,
int_formula_prop_eq_lemma,
bnot_wf,
not_wf,
uiff_transitivity,
eqtt_to_assert,
assert_of_eq_int,
iff_transitivity,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot,
exists_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
dependent_pairFormation,
lambdaEquality,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
productElimination,
sqequalRule,
independent_pairEquality,
applyEquality,
hypothesisEquality,
natural_numberEquality,
setElimination,
rename,
hypothesis,
independent_isectElimination,
independent_pairFormation,
functionExtensionality,
productEquality,
cumulativity,
isect_memberEquality,
voidElimination,
voidEquality,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_functionElimination,
axiomEquality,
dependent_set_memberEquality,
unionElimination,
int_eqEquality,
intEquality,
computeAll,
functionEquality,
universeEquality,
baseClosed,
equalityElimination,
impliesFunctionality
Latex:
\mforall{}[A,B:Type].
\mforall{}f0:A. \mforall{}g0:B. \mforall{}F:\mBbbN{} {}\mrightarrow{} A {}\mrightarrow{} B {}\mrightarrow{} A. \mforall{}G:\mBbbN{} {}\mrightarrow{} A {}\mrightarrow{} B {}\mrightarrow{} B.
\mexists{}f:\mBbbN{} {}\mrightarrow{} A
\mexists{}g:\mBbbN{} {}\mrightarrow{} B
(((f 0) = f0)
\mwedge{} ((g 0) = g0)
\mwedge{} (\mforall{}s:\mBbbN{}\msupplus{}. (((f s) = F[s - 1;f (s - 1);g (s - 1)]) \mwedge{} ((g s) = G[s - 1;f (s - 1);g (s - 1)]))))
Date html generated:
2018_05_21-PM-07_42_41
Last ObjectModification:
2017_07_26-PM-05_20_44
Theory : general
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